PART
I
: PHYSICS
SECTION 1 (Maximum Marks: 2l)
. This section contains SEVEN questions . Each question has FOUR options tAl, tBl, tcl and [D]. ONLY ONE of these four options is
I
correct
. .
For each question, darken the bubble corresponding to the correct option in the ORS For each que stion, marks
Marla : +3 Zero Marks : 0
Full
Negative
Q.l
Marks:
-1
will
be awarded in one
ofthe following categories:
If only the bubble corresponding to the correct option is darkened Ifnone ofthe bubbles is darkened In all other cases
A person measures the depth ofa well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is dI = 0.01 seconds and he measures the depth of the well to be L = 20 meters. Take the acceleration due to gravity g = 10 ms-2 and the velocity ofsound is 300 ms'r. Then the fractional error in the measurement, dl/l, is closest to
l{l
s%
lc)
3Yo
[D]
1%
0.2%
Space for rough work
V
,)/ -7
t \,r
\.u
'tt
'h
)
4
lr
..
Y 9"
,;Y
,
)
0
,t t/ f
a
3136
4t
n{ c)
@ C
0
e\
ao
t'
-t t
b
*}, 2
2\
,,,
<9
Q.2
Consider regular polygons with number of sides n = 3,4, 5...... as shown in the figure. The centcr of mass of all the polygons is at height h from the ground. They roll on a horizontal surface about the lcading vertex without slipping and sliding as depicted. The maximum increase in height of the locus ofthe center ofmass for each polygon is A. Then A depends on n and h as
I
.-'1
tAl
A= tt sin'(1)
rcr^=h(+-,)
rst ')
a=
h tan2
(a) \zn,t
tol a= irsin (a) Space for rough work
4t36
**2
Q.3
*a
R are shown in the figure. Let S be any point on the vector F. The distance between the points P and S is f lnl. fne general relation among vectors F, d na Three vectors F,
d
ii'
P
X\
i=a-F P
q o
x
bd tcl S=O-DF+bd tAl .i = (1-
Q.4
b1F +
t
tBl .i = (t - DF + bzd tDl S= @-DF+bd
A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is 3x10s times heavier than the Earth and is at a distance 2.5x10a times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is u" = 11.2 km s-1. The minimum initial velocity (ur) required for the rocket to be able to leave the Sun-Earth system is closest to (lgnore the rolation and revolution of the Earth and the presence of any other planel)
[A]
ur = 22 km
s-1
[B]
o, = 42kms-1
[C]
us
62 km
s-1
[D]
u5
=
=
72 km
s-l
Space for rough work
5/36
**2
Q.5
A symmetric star shaped conducting wire loop is carrying a steady state current I as shown in the hgure. The distance between the diametrically opposite vertices of the star is 4a. The magnitude ofthe magnetic field at the center ofthe loop is
I 4a
# 3tz-.V-31 tcl # st€- rl
tAt
lel tBt '-' 4ra 6tJ3 ' -
tol #otVs+rl Space for rough work
6/36
1t
Q.6
A photoelectric material having work-function @6 is illuminated with light of wavelength I (la \ *). The fastest photoelecron has a de Broglie wavelength ,ld. A change in oo,,
wavelength of the incident light by A.1 results A,Lo/A.l is proportional to
l^l Q.7
lBl
11d/1
in a change Llain ),6. Then the ratio
lcl
A?d/12
7a/ 7
11d112
/,
Consider an expanding sphere of instantaneous radius R whose total mass remains constant. The expansion is such that the instantaneous density p remains uniform throughout the volume. The rate of fractional change in densiry (1:, is constant. The velocity u ofany point on the surface ofthe expanding sphere is proportional to
lA)
R2/3
tBl
i
ICI
R
[D]
R,
Space for rough work
X -1.
?t)
o
L:-d -
Y1-*, ?r
h 2^ r/1 \z ,/,n /d
,,,\
hc-
----4
_
(
=/
62_
----.c'
2-
/' q-
12-
)
4?d= DT
E
[-c-
'..
l.')
2--/^
)
AC
b> ,r\t)
't t36
I
o tz-
u
^s'
a)\r>
z
a\,
g+,) t'.\'
)
SECTION 2 (Maximum Marks: 28)
. . . .
This section contains SEVEN questions Each question has FOUR options these four options is(are) correct
[A], [B], tcl and [D]. ONE OR MORE THAN ONE of
For each question, darken the bubble(s) corresponding to all the correct option(s) in the OIIS For each question, marks will be awarded in one ofthe following categories:
Full
Marl
Partial Zero
:
+4
Marl
Marks
Negative
:0
Marks : -2
If only
the bubble(s) corresponding to all the correct option(s)
is(are) darkened
For darkening a bubble corresponding to each correct option, provided NO incorrect option is darkened Ifnone ofthe bubbles is darkened In all other cases
For example, if IA], tC] and [D] are all the correct options for a question, darkening all these three will get +4 marks; darkening only [A] and [D] will get +2 marks; and darkening [A] and [B] will get -2 marks, as a wrong option is also darkened
Q.8
The instantaneous voltages at three terminals marked ,Y, Y and Z are given by Vx = Vo sin at'
= vosin(d +f,)ana v,= v"si^(,t+!). vv
An ideal voltmeter is configured to read rrns value of the potential difference between its terminals. It is cormected between points X and Y and then between Y and Z. The reading(s) of the voltrneter will be
l[l
Vffs =
Vo
lBl
v{i6 =
Yo
tcl VfrF =
Vo
tDl
It r/z
t;
.J,
independent ofthc choice ofthe two terminals
Space for rough work
8/36
Q.e
A wheel of radius R and mass M is placed at the bottom of a fixed step of height R as shown in the figure. A constant force is continuously applied on the surface of the wheel so that itjust climbs the step without slipping. Consider the torque r about an axis normal to the plane ofthe paper passing through the point Q. which ofthe following options is/are correct?
5
x
[A]
If the force is applied at point P tangentially then r decreases continuously wheel climbs
Ifthe force is applied normal to the circumference
--W tcl If
at point P then z is zero
the force is applied normal to the circumference at point X then
the forcc is applied tangentially at point .t then the step
Space for rough work
9136
as the
t
is constant
r + 0 but the wheel never climbs
Q.10 Two coherent monochromatic point sources .i, and s2 of waverength ,l = 600 nm are placed symmetrically on either side of the center of thi circle as shown. The sources are separated by a distance d 1.8 mm. This arrangement produces interference fringes visible as alternate bright and dark spots on the circumference of the circle. The angular separation between two consecutive bright spots is A0. Which of the following options is/are correct?
oP ,,
4
P2
&
J^ t 0
bo
[A]
A dark spot will be formed at the point
[B]
The angular separation between two consecuti from P1 to P2 along the first quadrant
bright spots decreases as we move
The^total number offringes produced between
p, and p2 in the first quadrant is close
tcl
p2
to 3000
At P, the order of the fringe will be maximum Space for rough work
{
t r I
\
6t
<.>
24
,,\ Id
) {
q
+( o^7 o
I
l0/36
o
I \o
Q.l
I A source of constant
voltage 7 is connected to a resistance R and two ideal inductors L, and 12 through a switch S as shown. There is no mutual inductance between the two inductors. The switch S is initially open. At t = 0, the switch is closed and current begins to flow. Which of the following options is/are correct?
5 R
s1l-2-
L1
L fr,2'' tAl After
a long time, the current through
tz will be f
tBl
At t = 0, the current through the resistance R is f,
tcl
The ratio ofthe currcnts through LlandL2 is hxed at all times
tDl
After a long trme. the current through
fr
wilf be f
I
f; (t > 0)
t';
Q.l2 A rigid uniform bar AB of length L is slipping from its vertical position on
a fiictionless made by the bar with the the angle of time, some inslant floor (as shown in the figure). At correct? is/are vertical is 0. Which of the following statements about its motion A e
o
ag
tAl
[D]
Instantaneous torque about the point in contact with the floor is proportional to sin 0 The midpoint of the bar
will fall vertically downward
When the bar makes an angle 0 with the vertical, the displacement of its midpoht from the initial position is proportional to (1 cos d)
-
The trajectory ofthe point A is a parabola
Space for rough work
ty36
Q.l3AuniformmagneticfieldBexistsintheregionbetweenr=0and:=f(region2inthe figure) pointing normally into the plane of the papcr. A particle with charge +Q and momentum p directed along x-axis enters region 2 from region I at point pr 0 = -R). Which ofthe following option(s) is/are correcr? Region 1
Region 2
Region 3
B
o P2
x
+QP (y
='R) 3Rt2
[A]
When the particle re-enters region I through the longest possible path in region 2, the magnitude of the change in its linear momentum between point p1 and the farthest point from y-axis is p/JZ
IW for e >'z1fi ,the particle will re-enter region
I
\/
_y tDl
For a fxed B, particles of same charge Q and same velocity u, the distance between the point P1 and the point of re-entry into region I is inversely proportional to the mass ofthe particle F6p
! = x J- , 15e particle will enter region 3 through the point p2 on.r-axis Space for rough work
12t76
charge +Q is placedjust outside an imaginary hemispherical surface of radius R as shown in the figure. Which of the following statements is/are correct?
Q.l4 A point
+Q
R
v
The component of the electric field normal to the flat surface is constant over the surface
9
tBl
Total flux through the curved and the flat surfaces is
[C]
The electric flux passing through the cun'ed surface of the hemisphere
_a
280
(,-iJ
The circumference ofthe flat surface is an equipotential
Space for rough work
t3/36
is
SECTION 3 (Maximum Marks: 12)
. . . . .
This section contains TWO paragraphs Based on each paragraph, there are
Each question has FOUR options is correct
TWO questions
[A], [B], [C], and [D]. ONLY ONE of these four options
For each question, darken the bubble corresponding to the correct option in the ORS For each question, marks will be awarded in one ofthe following categories:
Full
Marks : +3 Ifonly the bubble corresponding
Zero
Marks
: 0
to the correct option is darkened
In all other cases Space for rough work
14t36
**2
PARAGRAPH I Consider a simple RC circuit as shown in Figure I
process 1: In the circuit the switch S is closed at t = 0 and the capacitor is fully charged to >> RC). In the process some dissipation (E2) voltage 7s (i.e., charging continues for time occurs across the resistance R. The amount of energy finally stored in the fully charged capacitor is 86.
I
process 2: In a different process the voltage is first set to
f;
and maintained for a charging time
T>>RC. Then the voltage is raised to ?b without discharging the caPacitor and again maintained for a time T >> RC. The process is repeated one more time by raising the voltage to y0 and the capacitor is charged to the same final voltage 7s as in Process
1.
These two processes are depicted in Figure 2.
vo
Proaesr 1
5 2vJ3 Pro.eis
R
---+,\
2A
T>>RC
vo/3 C
f F
Q.l5 In Process
2f F
I
t
2
2, total energy dissipated across the resistance E2 is:
tAl
r, = z(lcq)
lcl
Eo=36Y;
Q.l6 In Process 1, the energy
tBl e,
=!$cvl)
cLD -
)rr"'
stored in the capacitor E6 and heat dissipated across resistance E2
are related by:
lAl
E, = 2s,
lC) E, = Po
lBl
E-
=!E^ 2"
Ec
= EolnZ
"
Space for rough work
15t36
**2
PARAGRAPH 2 one twirls a circular ring (of mass M and radius R) near the tip of one's finger as shown in Figure l. In the process the finger never loses contact with the inner rim ofthe iing. The finger traces out the surface of a cone, shown by the dotted tine. The radius of the path traced out by the point where the ring and the finger is in contact is r. The finger rotates with an angulir velocity aro. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is F and the acceleration due to gravity is g.
(:: ----I_
Figure I
Q.l7
Q.l8
The minimum value of
c,.rs
Figure 2 below which the ring will drop down is
rArffi B)ffi
l3s
tcl .J'fl(",
tDl
fr"
.Jll(R,
The total kinetic energy of the ring is
tAl )M4@-r)'
tBj :Ma|@ -
lC)
lDl Ma\@ -
MafiRz END OF PART
I
r)2
r)2
: PHYSICS
Space for rough work
t6t36
**2
PART
II : CHEMISTRY
SECTION
. .
I (Maximum Marks:
21)
This section contains SEVEN questions Each question has FOUR options
tAl, tBl, tcl and [D]. ONLY ONE of
these four options is
correct
. .
For each question, darken the bubble corresponding to the correct option in the ORS For each question, marks will be awarded in one of the followins categories:
Full Zero
Marks : +3 Marks : 0
Negative Marlcs:
Q.l9
-l
Ifonly
the bubble corresponding to the correct
option is darkened
Ifnone ofthe bubbles is darkened In all other cases
{L
For the following cell, z1 (s) | zh S o,
when the concentration AG (in J mol-r) is
(aql ll cu so,
of Zn2' is l0
1oq
1 |
cu
times the concentration
[F is Faraday constant; R is gas constant; T is temperatwe;
l[l -2.2F lcl l.rF
E
2303F.T
lDl
4{\
1s1
-
of
L
d
t"
Cu2' , the expression for
(cell) =1.1 2.2F
2.303RT + 1.lF
,r
vl
.f ,o/ K* \ I (! ,,v
"{,
.1
^3 ,,12
r,
v
Q.20 Thc order of basicity among the following compounds is NH
A*r, Hrc I
[A] IV>II>III>I
[c] It>I>IV>III
1'
NH"
N\,,NH II
N..ZN
HzN
III
'. -Y>(w
A; tV
lBl I>IV>III>II Space for rough work
17136
4,,t .) \t :lt ,)A
'>rt'rrt
.,((,' .
)
a
,,,DG
,,,
\.i-
D
D
(
,r:r,
X
Q.2l
,f ';
Pure water freezcs at 273 K and 1 bar. The addition of 34.5 g of ethanol to 500 g of water changes the freezing point of the solution. Use the freezing point depression constant of watcr as 2 K kg mol-'. The figures shown below represent plots ofvapour pressure (V.P.) versus temperature (T). [molecular weight of ethanol is 46 g mol-]] Among the following, the option representing change in the freezing point is
tAl
0
tBl
N.
1 I
I
c
llater
+ Ethanot
+l-
270
273
T/K
271
-------+
tcl
T/K+
tDl
1
0
273
I
t
A:
\'i +o,h.not 211
,2,
213
270
T/K """'+
273
T/K
-------r
x
ST
Q.22
T\e standard T=298Kare
state Gibbs free energies
of formation of C(graphite) and C(diarnond)
at
ArG'[C(graphite)]= 0 kJ mol-r ArG'[C(diamond)] = 2.9 kJ mol-r. The standard state means that the pressure should be I bar, and substance should be pure at a given temperature. The conversion of graphite I C(graphite) ] to diamond IC(diamond)]
by 2x10-6 m3 mol-r. If C(graphite) is converted to C(diamond) isothermally at T = 298 I! the pressure at which C(graphite) is in equilibrium with
reduces its volume C(diamond), is
[Useful information: I
[A]
s8001 bar
J= l
[B]
kg m2 s-2; 1450 bar
I Pa= 1kg
rs
14501
Space for rough work
l8/36
2;
I
bar
bar
=
105 Pa]
[D]
29001 bar
Q.23 The order of the oxidation state of the phosphorus atom HrP: Oo is
in H;POz, H:POr, H:PO:,
,6
H:Poz > H:Po: > FLP:oo >
tn
tY
H3POa
H:Por Bl ffop J'
> FLPIOc >HrPOr> HrPOz
and
o
,?nor,
^rlor,
,#ou
HrPO:> H:POz > H:POr> HrPzOr
Q.24 Which of the following combination will produce H2 gas?
[A] Cu metal and conc. HNO3 [B] Au metal and NaCN(aq) in the presence of air --A{ Zn metal and NaOH(ad [D] Fe metal and conc. HNO3
r)
c)
tl
(
Oll-
( ------.\ 0___:Y"
l
ot4
(
6
oH
o
Q.25 The major product ofthe following reaction is
,rr-t!rr
OH i) NaNO2, HCl, OoC
NHz
o
ii) aq. NaOH
rl
tAl
ou- (
tBl OH
,l-
N=N C
tDl O Na+
N N2Cl
Space for rough work
t9t36
NOH
()' r-'l
r.l
SECTION 2 (Maximum Marks: 28)
. .
This section contains SEVEN questions Each question has FOUR options these four options is(are) correct
[A], [B], [C] and [D]. ONE OR MORE TIIAN ONE of
.
For each question, darken the bubblc(s) corresponding to all the correct option(s) in the ORS
.
For each question, marks will be awarded in one of the followine catesories
Full
.
Marl
:+4
Partial Marks
:+l
Zero Marl
:0 .,
If only the bubble(s)
conesponding to all the correct option(s)
is(are) darkened For darkening a bubble corresponding to each correct option, provided NO incorrect option is darkened Ifnone ofthe bubbles is darkened In all other cases
if tAl, tC] and [D] are all the correct options for a qucstion, darkening all these tbree will get +4 marks; darkening only [A] and [D] will get +2 marks; and darkening [A] and [B] will get -2 marks, as a wrong option is also darkened For example,
Q.26 In a bimolecular reaction, the stcric factor P was experimentally determined to be 4.5. The correct option(s) among the following is(are)
[A]
The activation energy of the reaction is unaffected by the value ofthe steric factor
]
The value of frequency factor predicted by Arrhenius equation is higher than that determined experimentally
l
Silce P = 4.5, the reaction will not proceed unless an effective catalyst is used
tDl
Experimentally determincd value of frequency factor is higher than that predicted by Arrhenius equation Space for rough work
20t36
*),2
1N
tv'
l.^.
F\)
Q.27 For a reaction taking place in a container in equilibrium with its surroundings, the effect o temperature on its equilibrium constant I( in terms of change in entropy is described by
[A]
With increase in temperature, the value of .l( for exothermic reaction because the entropy change of the system is positive \/
[B]
With increase in temperature, the value because the entropy change
lcx
of K for endothermic reaction increases ofthe system is negative V,"
of Kfor exothermic reaction decreases ofthe surroundings decreases
With increase in temperature, the value because favourable change in entropy
D
decreases
With increase in tcmperature, the value of r( for endothermic reaction
increases
because unfavourablc change in entropy of the surroundings decreases
Q.28 The option(s) with only amphoteric oxides is(are)
lAl
CrzO:, CrO, SnO, PbO
[c]
No, Bzo:, Pbo, Snoz
B
:, BeO, SnO, SnOz ZnO, AlzOz, PbO, PbOu
Space for rough work
2y36
**2
Q.29 Among the following, the conect statement(s) is(are)
]
BHI has the three-centre two-electron bonds in its dimeric structure
[B]
The Lewis acidity of BCl3 is greater than that of
[C] [D]
A(CH3)3 has the three-centre two-elechon bonds in its dimeric structure
AlCl:
AICI3 has the three-centre two-electron bonds in its dimeric structure
Q.30 Compounds P and R upon ozonolysis produce Q and S, respectively. The molecular formula ofQ and S is CrHsO. Q undergoes Cannizzarc reaction but not haloform reaction, whereas S undcrgoes haloform reaction but not Cannizzaro reaction.
(i) *
D
orlCHzCl:
ii) ZDI{.O
(ii)
-
i) o3lcH2cl2
R
ii)
A (csHso)
znlHzo
S
(csHEo)
The option(s) with suitable combination of P and R, respectively, is(are)
tAl Hrc
tBl and
and H3
tcl
tDl cHr
H:C
and
and
HtC
Space for rough work
22136
**2
Q.3l
The correct statement(s) about surface properties is(are)
[A]
J' [C]
Brownian motion of colloidal particles does not depend on the size of the particles but depends on viscosity ofthe solution The critical temperatures of ethane and nitrogen are 563 K and 126 K, respectively. The adsorption of ethane will be more than that of nitrogen on same amount of activated charcoal at a givcn temperature
Cloud is an emulsion type of colloid in which liquid is dispersed phase and gas is dispersion medium
lD) Adsorption is accompanied by decrease in enthalpy
and decrease in entropy of the
system
Q.32
For the following compounds, the correct statement(s) with respect to nucleophilic substitution reactions is(are) Br
I
* o ''"-E;i c{;' [trITV
II follow SN2 mechamsm I and III follow SNI mechanism
[A]
I
tcl
Compound
tDl
The order ofreactivity for
and
IV undergoes inversion of
confi guration
I, III and IV is: IV > I > III
Space for rough work
23t36
SECTION 3 (Maximum Marks: 12)
. . .
This section contains TWO paragraphs Based on each paragraph, there are
Each question has FOUR options is correct
. .
TWO questions
[A], [B], [C], and [D]. ONLY ONE of these four options
For each question, darken the bubble corresponding to the correct option in the ORS For each que stion, marks
will
: +3 If only the bubble Marks : 0 In all other cases
Full Marlcs Zero
be awarded in one
ofthe following catesories
corresponding to the correct option is darkened
PARAGRAPH
I
Upon heating KCIO: in the presence of catalytic amount of MnO2, a gas W is formed. Excess w reacts with white phosphorus to give X. The reaction of X with pure HNo3 gives
amount of
Y
and
Z.
Q.33 Y and Z are, respectively
[A]
N2Oa and HPO3
[C]
NzO: and H;POr
-----[6iNzos
and HPo:
zOr and H:PO:
Q.34 W and X are, respectively
[A]
03 and PaOls Oz and PrOro
(r, [D]
o,andPao6 O3 and PaO6
Space for roug work
24t36
**2
PARAGRAPH
2
The reaction of compound P with CHsMgBr (excess) in (C2H5)2O followed by addition of HzO gives Q. The compound Q on treatment with H2SO4 at OoC gives R. The reaction of R with CH:COCI in the presence of anlydrous AlCl: in CHzClz followed by treatment with HzO produces compound S. [Et in compound P is ethyl group] (H
a
R
S
P
Q.35 The reactions, Q to R and R to S, are
[A] [B]
Friedel-Crafts alkylation and Friedel-Crafts acylation Friedel-Crafts alkylation, dehydration and Friedel-Crafts acylation Aromatic sulfonation and Friedel-Crafts acylation
tDl
Dehydration and Friedel-Craft
s
acylation
Q.36 The product S is
tBl
tAl
HO3S H3
(H3C)3C
(H3C)3C
coc H 3\,
cocH3
tDl
(H3C)sC
H3
(H3ChC
coc H3 CH:
cocH3 END OF PART
II : CHEMISTRY
Space for rough work
25136
**2
PART
III : MATHEMATICS
SECTION
. .
This section contains SEVEN questions Each question has FOUR options
correct
. .
I (Maximum Marks: 2l)
tAl, tBl, tcl
and [D]. ONLY ONE of these four options is
For each question, darken the bubble corresponding to the correct option in the ORS For each question, marks will be awarded in one
Full
Marl
:
Zero Marks 0 Negative Marks: -l
of the fo
catcsories:
If only the bubble corresponding to the correct option is darkened Ifnone ofthe bubbles is darkened ln all other cases
Q.37 Let S = {L,2,3, ... ,9}. For k = 7,2, ...,5, let N2 be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N1 + N2 + N3 + IV4 + Ns =
lAl 125
lB) 2s2
[c] 210
[D]
126
Q.38 Let O be the origin and let PQR be an arbitrary triangle. The point S is such that
oF.od+ ort.6= ofr.oi+ od oi= od.oi+ oF.oi Then the triangle PQR has S as its
[A]
orthocenter
[C]
circumcentre
0 ,)
centroid
[D]
2
incentre
D,
Space for rough work
&p-
0
t
10P t
o{ .o
or-
o
o
05 26/36
,,], 2
Q.39
If f:R
-+ IR
is a rwice differentiable function such that
r()) =).to
tBl i
tct 0< f'(D3: planes 2x
o(,
and
= 1,then
tAl /'(1) < 0
Q.40 The equation
f"(x)> 0 for all r€R,
f'(t)
>
1
of the plane passing tkough the point (1, 1, 1) and perpendicular to 2z = 5 and 3: - 6y - 2z = 7, is
t y-
-2y *
74x12y-LSz=7
[B]
74xtZyt75z=37
[D] -14r + 2y + l5z = 3
14x
752
=
27
Space for rough work
.L
4L I
o
lqd-'
2p -l9c
4-J=
2-'
-L
3-€
-2-
o o
-12-
L
rq
le'
) --J
+r;
27t36
the
t
Lt
l-6
Ftz-3 )
**2
nl,
I
Q.4l How many
3
x 3 matrices
M with entries fiom {0,
1, 2} are
there, for which the sum of the
diagonal entries of Mr M is 5?
[A]
[B]
162
lcl
135
Q.a2 If y = y(x) satisfies the differential equation
and
y(0)
tAl
:
8,li
e
tl7,
then
+li
dv=
4+
198
_L
9
+"1,
x>0
dx,
y(256) =
e
[c]
3
a
tBt
,11
tDl
80
Q.43 Three randomly chosen nonnegative integers x,y and x + y + z = 10. Then the probability that z is even, is .rAt ,
lDl
126
z
16
are found to satisry the equation
tq-
1
"2
6 11
Space for rough work
1
k
,
a,
4
\P
x{*
),"--. Lb
^{"
qN
/-
q
u
o
)6
++
L1
l-
I
,.,-A
+
I 28t36
,'-:J
c_,
r@
& 1
q+
,
7'r''
4+ ).), 2
qF.
SECTION 2 (Maximum Marks: 28)
. .
This section contains SEVEN questions Each question has FOUR options
[A], [B], [C] and [D]. ONE OR MORE THAN ONE of
these four options is(are) correct
d
.
For each question, darken the bubble(s) conesponding to all the correct option(s) in the ORS
.
For each question, marks will be awarded in one ofthe following categories:
(9
+4
Full Marks
"rt \\y \/v .
Partial Marks
+1
Zero Marks Negative Marl
0 -2
If only
the bubble(s) corresponding to all the correct option(s)
is(are) darkened For darkening a bubble corresponding to each correct option, provided NO incorrect option is darkened Ifnonc ofthe bubbles is darkened ln all other cascs
For exarnple, if tA], tC] and [D] are all the correct options for a question, dalkening all these tkee wili get +4 marks; darkening only [A] and [D] will get +2 marks; and darkening [A] and [B] will get -2 marks, as a wrong option is also darkened
Q.44
Ifthelioex=adividestheareaofregionR:{(x,y)€lR2:23
Q.4s
l$ )
l-Bl
[C] dr+4a2-7=o
lD)
If 9(x) = fhG')
s'(-i)
Zaa
-
4d2 +
1-
-t/ rl
-- 0
= -z1r
4'H=-," tDt
s,(-:)
=
^
zTr
Space for rough work
, /L s,v) Cu"'t' ) czaz't
29136
/r.: /N
sin-r1t) dt, then
b1 s'O=zn 1c:l
l'
0
)-L>
Y4 x7
Q.46 Let f
(x)
."r (+_J
x+
for
1. rhen
lim,-r /(x) does not exist
lim,-1*
[C] lim,-r. f (x) = 0 Q.47
If /:
R
---r
,f (0) =
R is a differentiable tunction such that f,(x) > Zf (x) for all z € R,
in (0,
\-/
oo)
,p{ trrr>ez,in (0,oo) lOl f (x) is decreasing in (0,o)
Q48 Let a and p be nonzero real numbers such that Z(cos p which of the following is/are tue?
v
o tct Jst"(i)+t,n(4)=o
/
and
1, ttren
19 f@) is increasing
ta,
does not exist
[D] lim,-r- /(x) = a
IN f!)
tAl
/(r)
(;)
+ r's t"n (a) =
1e1 t,n
- cosa) + cosacosp
=
1. Then
(i) - V:t (f) = o "
tot /:t""(,g)-""(9=o l.
Space for rough work
Ir .h
I
,*,!c
l
4
t
ja
3
u)
l.J
(tn)
,tl t
//
q
0/36
h
t,'
-h
t+
lr t-r'
{
{
/'rh I
\/
I
5'f/ o
\-(
tr
Itu) kh
I
I
tr
lr
**2
r.l-.)
fi- t-"'l
"1"1
,.-)-
). Y itr\
4\
!a
cos(2r) cos(2:) sin(Zr)
- / cos I
- cos.I cos .r sin: sinr
Q.aercf@)=
sin
lel f'(x) = El f (x) attains its minimum at /
0 at exactly three points in
/(x)
attains its maximum at
=
0
x=
0
[x+l?
, then
1
I^ k\
(-2, n)
tl
\
"L
( CV
1,
f '(x) = 0 at more than tfuee points in (-2, n)
)
\ -h L .@|il t"
o.sorrr=)
98
l.k+1 k+1 dr rk x(r+1)
then
lAl /< 50
I
q1'
I > log,99 50
il\
/)
+
'u'L
4\
6(
d_
w
I
+* A ^rA-o-S*
Oz
I
dL'r\ -(2 CTJ
\
t'^ er|
y'e, ?t ='/
w , 't)2^ t/L \2 ?,( I,L
z
2 2/> x-a-9
I
)
d, ht
O
<^
I
Sioxsin Cr9t u9 Zl.-.-
-7-o''9
3tt36
.,|
,fr ZCO
.
(0
Oe3"'c--
vlJ
i
I
nt/
5fn2)X+ ZCe-e X- -.-'sr\"<-
n1<-
41
L
C)
-
.1
Space for rough work
, tu8 Z 71-'
dft't
td+So
49
[C] /
/:I
^,[
\\
1
-
49
t-
\(
,^)
,2*1+t/ )
^.oZ-L
,"5 a'Y<'-{"'
za-* zsua't'-= lF'' o ,,* +l= c.2Lr'' 2h ,,7 'Zc>< ,t
o
a
2
,1
o
-2-
{
I
r
L
SECTION 3 (Maximum Marks: 12)
. This section contains TWO paragraphs . Based on cach paragraph, there are TWO questions . Each question has FOUR options tel, tBl, [C], and [D]. ONLY ONE of these four options is correct
.
For cach question, darken the bubble conesponding to thc correct option in the ORS
.
For each question, marks
will
be awarded in one of the
Full
Marks : +3 If only the bubble corresponding
Zero
Marks
: 0
cateqories: to the correct option is darkened
In all other cases
PARAGRAPH I Let O be the origin, and
ofa Q.sr
I
oixoi
[A]
07, O?,
OZ
t" three unit vectors in the directions ofthe sides Qf, RF,
P R
l--
sin(P + R)
[B]
[C]
sin 2R
sin(P +
Q) [D] sin(Q + R)
Q.52 If the triangle PQR varies, then the minimum value of cos(P + Q) *cos(Q * R) +cos(R + p) is
rAl
-;
[Bt
:
rcl
-;
rDl
:
Space for rough work
?
c
/')
(u A
32t36
o
(L
--.>
*,.2
:-i.
PARAGRAPH 2
,9 Let
p, qbe integers and let a, p be the roots of the equation, x2 -x- 1= 0' where
a + P.For n = 0,7,2,... ,let
o.n
= pa"
*
48".
FACT: If a and b are rational numbers and a + b^15 -Q.53 412
O
then a
=
0
=
b.
=
[A] Q.54 If ca
tAl
2411
*
f ar- an
a16
fC) af !2a1s
[D]
411
tcl
lDl
12
+
a1o
= zB,thenP + 2q =
21
[B]
14
7
END OFTHE QUESTION PAPER Space ,or rough work
,,u
(tru
,
h
?h ;Ax1/ r h (
A
+
\4./ a
I ql
.//
\
F
h(\7 l"
v\)
?
u(
+{t a /
\+G
_€
l+€ ,tLg rfL
?
L4 S 6r
{+ 33136
ZD
t5
.(s
ILt"
-fr **2
